Improved asymptotic bounds for codes using distinguished divisors of global function fields

Abstract

For a prime power $q$, let ${\alpha }_q$ be the standard function in the asymptotic theory of codes, that is, ${\alpha }_q(\delta )$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance $\delta$ of $q$-ary codes. In recent years the Tsfasman-Vlăduţ-Zink lower bound on ${\alpha }_q(\delta )$ was improved by Elkies, Xing, and Niederreiter and Özbudak. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. We also show improved lower bounds on the corresponding function ${\alpha }_q^{\mathrm{lin}}$ for linear codes.